This dependency on the slope was studied explicitly in [1]} for the moduli space of framed representations of a toric quiver, and this was related to the ambiguity in the refined topological vertex of [2]}. In this case, there is a two dimensional torus invariant acting on the moduli space of framed representations, scaling the arrows of the quiver by leaving the potential invariant, hence the space of slopes is \(\mathbb {P}^1_\mathbb {R}\) . The fixed points can be described as molten crystals from [3]}. In [1]}, it was established that there is a wall and chamber structure on the space of slopes, with the generating functions of framed invariants being constant in a chamber and jumping at a wall, the wall corresponding to slopes where the weight of an elementary cycle of the quiver becomes attracting or repelling. This is quite strange at first sight, because inside a given chamber the cohomological weight of a given molten crystal does changes on many walls, but the final result does not change: those walls are 'invisible'. In [5]} it was established that the attracting variety is the subspace of representations where the cycles with repelling weights are nilpotent, hence the attracting variety changes exactly on the walls defined in [1]}, i.e. (REF ) give an explanation of this wall and chamber structure. Moreover, using a nilpotent/invertible decomposition for unframed representation and a wall-crossing relation between framed and unframed invariants, [5]} obtained the full framed generating series by multiplying the one obtained by localization by a generating series of framed invariants where some cycles are imposed to be invertible. The latter is simple to compute and has a universal closed formula for all toric quivers. Notice that in this case the moduli space is the critical locus of the potential of the quiver, hence one does not need all the subtleties of the gluing, i.e. one needs only proposition REF .
